\(\int \frac {(d+e x^2)^{5/2}}{a-c x^4} \, dx\) [342]

Optimal result

Integrand size = 22, antiderivative size = 199 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=-\frac {e^2 x \sqrt {d+e x^2}}{2 c}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c^{3/2}}-\frac {5 d e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c^{3/2}} \] Output:

-1/2*e^2*x*(e*x^2+d)^(1/2)/c+1/2*(c^(1/2)*d-a^(1/2)*e)^(5/2)*arctan((c^(1/ 
2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/c^(3/2)-5/2*d*e^( 
3/2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/c+1/2*(c^(1/2)*d+a^(1/2)*e)^(5/2)* 
arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/c^( 
3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.24 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.95 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=-\frac {e^{3/2} \left (\sqrt {e} x \sqrt {d+e x^2}-5 d \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+\text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {3 c d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+a d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+10 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+3 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{2 c} \] Input:

Integrate[(d + e*x^2)^(5/2)/(a - c*x^4),x]
 

Output:

-1/2*(e^(3/2)*(Sqrt[e]*x*Sqrt[d + e*x^2] - 5*d*Log[-(Sqrt[e]*x) + Sqrt[d + 
 e*x^2]] + RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 4*c 
*d*#1^3 + c*#1^4 & , (3*c*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2 
] - #1] + a*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 
2*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 10*a*d*e^ 
2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c*d^2*Log[d + 
 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + a*e^2*Log[d + 2*e*x^2 
- 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 
 + 3*c*d*#1^2 - c*#1^3) & ]))/c
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(428\) vs. \(2(199)=398\).

Time = 1.13 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1489, 27, 318, 25, 403, 25, 398, 224, 219, 291, 218, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx\)

\(\Big \downarrow \) 1489

\(\displaystyle \frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}dx}{2 \sqrt {a}}+\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}dx}{2 \sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {a}-\sqrt {c} x^2}dx}{2 \sqrt {a}}+\frac {\int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} x^2+\sqrt {a}}dx}{2 \sqrt {a}}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d-4 \sqrt {a} e\right ) x^2+d \left (4 \sqrt {c} d-\sqrt {a} e\right )\right )}{\sqrt {c} x^2+\sqrt {a}}dx}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {-\frac {\int -\frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d+4 \sqrt {a} e\right ) x^2+d \left (4 \sqrt {c} d+\sqrt {a} e\right )\right )}{\sqrt {a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d-4 \sqrt {a} e\right ) x^2+d \left (4 \sqrt {c} d-\sqrt {a} e\right )\right )}{\sqrt {c} x^2+\sqrt {a}}dx}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d+4 \sqrt {a} e\right ) x^2+d \left (4 \sqrt {c} d+\sqrt {a} e\right )\right )}{\sqrt {a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {\int \frac {e \left (15 c d^2-20 \sqrt {a} \sqrt {c} e d+8 a e^2\right ) x^2+d \left (8 c d^2-9 \sqrt {a} \sqrt {c} e d+4 a e^2\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {-\frac {\int -\frac {e \left (15 c d^2+20 \sqrt {a} \sqrt {c} e d+8 a e^2\right ) x^2+d \left (8 c d^2+9 \sqrt {a} \sqrt {c} e d+4 a e^2\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {e \left (15 c d^2-20 \sqrt {a} \sqrt {c} e d+8 a e^2\right ) x^2+d \left (8 c d^2-9 \sqrt {a} \sqrt {c} e d+4 a e^2\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\int \frac {e \left (15 c d^2+20 \sqrt {a} \sqrt {c} e d+8 a e^2\right ) x^2+d \left (8 c d^2+9 \sqrt {a} \sqrt {c} e d+4 a e^2\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {e \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {e \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {8 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {\frac {8 \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {a} \sqrt {c} d e+8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {a}}\)

Input:

Int[(d + e*x^2)^(5/2)/(a - c*x^4),x]
 

Output:

((e*x*(d + e*x^2)^(3/2))/(4*Sqrt[c]) + ((e*(7*d - (4*Sqrt[a]*e)/Sqrt[c])*x 
*Sqrt[d + e*x^2])/2 + ((8*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTan[(Sqrt[Sqrt[ 
c]*d - Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)*Sqrt[c]) + (Sqrt 
[e]*(15*c*d^2 - 20*Sqrt[a]*Sqrt[c]*d*e + 8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt 
[d + e*x^2]])/Sqrt[c])/(2*Sqrt[c]))/(4*Sqrt[c]))/(2*Sqrt[a]) + (-1/4*(e*x* 
(d + e*x^2)^(3/2))/Sqrt[c] + (-1/2*(e*(7*d + (4*Sqrt[a]*e)/Sqrt[c])*x*Sqrt 
[d + e*x^2]) + (-((Sqrt[e]*(15*c*d^2 + 20*Sqrt[a]*Sqrt[c]*d*e + 8*a*e^2)*A 
rcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c]) + (8*(Sqrt[c]*d + Sqrt[a]*e) 
^(5/2)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])]) 
/(a^(1/4)*Sqrt[c]))/(2*Sqrt[c]))/(4*Sqrt[c]))/(2*Sqrt[a])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(143)=286\).

Time = 0.58 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48

method	result	size
pseudoelliptic	\(\frac {\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, d \left (e^{3} a^{2}+3 d^{2} e a c -3 \sqrt {d^{2} a c}\, a \,e^{2}-\sqrt {d^{2} a c}\, c \,d^{2}\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (d \left (\left (3 a \,e^{2}+c \,d^{2}\right ) \sqrt {d^{2} a c}+a e \left (a \,e^{2}+3 c \,d^{2}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )-\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\, \left (\sqrt {e \,x^{2}+d}\, e^{2} x +5 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) d \,e^{\frac {3}{2}}\right )\right )}{2 \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, c}\)	\(295\)
risch	\(-\frac {e^{2} x \sqrt {e \,x^{2}+d}}{2 c}-\frac {-\frac {\left (c^{\frac {3}{2}} d^{3}-a^{\frac {3}{2}} e^{3}+3 a \sqrt {c}\, d \,e^{2}-3 \sqrt {a}\, c \,d^{2} e \right ) \left (-\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}\right )}{\sqrt {a}\, \sqrt {c}}+\frac {\left (3 \sqrt {a}\, c \,d^{2} e +a^{\frac {3}{2}} e^{3}+3 a \sqrt {c}\, d \,e^{2}+c^{\frac {3}{2}} d^{3}\right ) \left (-\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}\right )}{\sqrt {a}\, \sqrt {c}}+5 d \,e^{\frac {3}{2}} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 c}\)	\(902\)
default	\(\text {Expression too large to display}\)	\(4738\)

Input:

int((e*x^2+d)^(5/2)/(-c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

1/2/(d^2*a*c)^(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)/((-a*e+(d^2*a*c)^(1/2) 
)*a)^(1/2)*(((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*d*(e^3*a^2+3*d^2*e*a*c-3*(d^2* 
a*c)^(1/2)*a*e^2-(d^2*a*c)^(1/2)*c*d^2)*arctan((e*x^2+d)^(1/2)/x*a/((-a*e+ 
(d^2*a*c)^(1/2))*a)^(1/2))+((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(d*((3*a*e^2+c 
*d^2)*(d^2*a*c)^(1/2)+a*e*(a*e^2+3*c*d^2))*arctanh((e*x^2+d)^(1/2)/x*a/((a 
*e+(d^2*a*c)^(1/2))*a)^(1/2))-((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(d^2*a*c)^(1 
/2)*((e*x^2+d)^(1/2)*e^2*x+5*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))*d*e^(3/2)) 
))/c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2299 vs. \(2 (143) = 286\).

Time = 12.90 (sec) , antiderivative size = 4605, normalized size of antiderivative = 23.14 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(5/2)/(-c*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=- \int \frac {d^{2} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx - \int \frac {e^{2} x^{4} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx - \int \frac {2 d e x^{2} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx \] Input:

integrate((e*x**2+d)**(5/2)/(-c*x**4+a),x)
 

Output:

-Integral(d**2*sqrt(d + e*x**2)/(-a + c*x**4), x) - Integral(e**2*x**4*sqr 
t(d + e*x**2)/(-a + c*x**4), x) - Integral(2*d*e*x**2*sqrt(d + e*x**2)/(-a 
 + c*x**4), x)
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{c x^{4} - a} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)/(-c*x^4+a),x, algorithm="maxima")
 

Output:

-integrate((e*x^2 + d)^(5/2)/(c*x^4 - a), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x^2+d)^(5/2)/(-c*x^4+a),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{a-c\,x^4} \,d x \] Input:

int((d + e*x^2)^(5/2)/(a - c*x^4),x)
 

Output:

int((d + e*x^2)^(5/2)/(a - c*x^4), x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a-c x^4} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{-c \,x^{4}+a}d x \right ) d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{-c \,x^{4}+a}d x \right ) e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{-c \,x^{4}+a}d x \right ) d e \] Input:

int((e*x^2+d)^(5/2)/(-c*x^4+a),x)
 

Output:

int(sqrt(d + e*x**2)/(a - c*x**4),x)*d**2 + int((sqrt(d + e*x**2)*x**4)/(a 
 - c*x**4),x)*e**2 + 2*int((sqrt(d + e*x**2)*x**2)/(a - c*x**4),x)*d*e